# 4.5 Add and subtract fractions with different denominators  (Page 4/7)

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Add: $-\frac{13}{42}+\frac{17}{35}.$

$\frac{37}{210}$

Add: $-\frac{19}{24}+\frac{17}{32}.$

$-\frac{25}{96}$

In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

Add: $\frac{3}{5}+\frac{x}{8}.$

## Solution

The fractions have different denominators.

 $\frac{3}{5}+\frac{x}{8}$ Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. $\frac{24}{40}+\frac{5x}{8}$ Add. $\frac{24\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}5x}{40}$

We cannot add $24$ and $5x$ since they are not like terms, so we cannot simplify the expression any further.

Add: $\frac{y}{6}+\frac{7}{9}.$

$\frac{3y+14}{18}$

Add: $\frac{x}{6}+\frac{7}{15}.$

$\frac{5x+14}{30}$

## Identify and use fraction operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

## Summary of fraction operations

Fraction multiplication: Multiply the numerators and multiply the denominators.

$\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}$

Fraction division: Multiply the first fraction by the reciprocal of the second.

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$

Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$

Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

$\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}$

Simplify:

1. $-\frac{1}{4}+\frac{1}{6}$
2. $-\frac{1}{4}÷\frac{1}{6}$

## Solution

First we ask ourselves, “What is the operation?”

Do the fractions have a common denominator? No.

 $-\frac{1}{4}+\frac{1}{6}$ Find the LCD. Rewrite each fraction as an equivalent fraction with the LCD. Simplify the numerators and denominators. $-\frac{3}{12}+\frac{2}{12}$ Add the numerators and place the sum over the common denominator. $-\frac{1}{12}$ Check to see if the answer can be simplified. It cannot.

The operation is division. We do not need a common denominator.

 $-\frac{1}{4}÷\frac{1}{6}$ To divide fractions, multiply the first fraction by the reciprocal of the second. $-\frac{1}{4}·\frac{6}{1}$ Multiply. $-\frac{6}{4}$ Simplify. $-\frac{3}{2}$

Simplify each expression:

1. $-\frac{3}{4}-\frac{1}{6}$
2. $-\frac{3}{4}·\frac{1}{6}$

1. $-\frac{11}{12}$
2. $-\frac{1}{8}$

Simplify each expression:

1. $\frac{5}{6}÷\left(-\frac{1}{4}\right)$
2. $\frac{5}{6}-\left(-\frac{1}{4}\right)$

1. $-\frac{10}{3}$
2. $\frac{13}{12}$

Simplify:

1. $\frac{5}{x}-\frac{3}{10}$
2. $\frac{5}{x}·\frac{3}{10}$

## Solution

The operation is subtraction. The fractions do not have a common denominator.

 $\frac{5x}{6}-\frac{3}{10}$ Rewrite each fraction as an equivalent fraction with the LCD, 30. $\frac{5x·5}{6·5}-\frac{3·3}{10·3}$ $\frac{25x}{30}-\frac{9}{30}$ Subtract the numerators and place the difference over the common denominator. $\frac{25x-9}{30}$

The operation is multiplication; no need for a common denominator.

 $\frac{5x}{6}·\frac{3}{10}$ To multiply fractions, multiply the numerators and multiply the denominators. $\frac{5x·3}{6·10}$ Rewrite, showing common factors. $\frac{\overline{)5}·x·\overline{)3}}{2·\overline{)3}·2·\overline{)5}}$ Remove common factors to simplify. $\frac{x}{4}$

Simplify:

1. $\frac{3a}{4}-\frac{8}{9}$
2. $\frac{3a}{4}·\frac{8}{9}$

1. $\frac{27}{a}$
2. $\frac{2}{a}$

Simplify:

1. $\frac{4k}{5}+\frac{5}{6}$
2. $\frac{4k}{5}÷\frac{5}{6}$

1. $\frac{24}{k}$
2. $\frac{24}{k}$

## Use the order of operations to simplify complex fractions

In Multiply and Divide Mixed Numbers and Complex Fractions , we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

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